Spring 2016
Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall II, unless noted otherwise. Refreshments are served at 11:30 am. For questions about the seminar schedule, please contact Yassine Boubendir.
Date: April 29, 2016
Speaker: Michael Zabarankin
Department of Mathematical Sciences,
Stevens Institute of Technology
Title: "Analytical Solution for Spheroidal Drop Under Axisymmetric Linearized Boundary Conditions"
Abstract:
A liquid spheroidal drop freely suspended in another fluid is considered under arbitrary axisymmetric boundary conditions, which are linearized with respect to the velocity field and can result, in particular, from axisymmetric external flow and electric field being applied either separately or in combination. All nonlinear effects including inertia and surface charge convection are assumed to be negligible, whereas the drop and the ambient fluid are assumed to be leaky dielectrics and to have different viscosities. Central to the analysis are the reformulated stress boundary conditions and representation of the velocity field in and out the drop in terms of non-Stokes stream functions. In the prolate spheroidal coordinates, the stream functions are expanded into infinite series of spheroidal harmonics, and the reformulated velocity and stress boundary conditions yield a first-order difference equation for the series coefficients, which admits an exact non-recursive solution. "Steady" spheroidal drops then correspond to minimums of a kinematic condition error that admits a simple efficient approximation. Under the simultaneous presence of aligned linear flow and uniform electric field with corresponding capillary numbers Ca and Ca_E, a spherical drop is stationary when Ca = k Ca_E and becomes prolate/oblate when Ca > k Ca_E (Ca < k Ca_E), where k is proportional to the Taylor discriminating function and depends on ratios of viscosities, dielectric constants and electric conductivities of the two phases. A spheroidal drop is "steady” when Ca = k_1 Ca_E + k_2, where k_1 and k_2 depend on spheroid's axes ratio d and approach k and 0, respectively, as d—>1. The results show that when the Taylor deformation parameter D is in the range [-0.5,0.4], this relationship can be used for finding any of the three Ca, Ca_E and D when the other two are given.